I would like to know which one of these expressions is the most correct and why?
The car of Google
When I refer to the driverless car Google has invented.
English Language & Usage Stack Exchange is a question and answer site for linguists, etymologists, and serious English language enthusiasts. It only takes a minute to sign up.Sign up to join this community
They are both correct, but most people would say Google's car in everyday speech.
This is usually used with particular things which are possessed by named people/companies/countries etc
Thus: "Let's go in John's car" but never "Let's go in the car of John."
However, you could use it for emphasis: "This car of John's is going to cost him a fortune."
The of construction is used in more general terms.
The climate of the world is changing
There are also various traditional/historic uses:
The Tower of London
The Leaning Tower of Pisa
The soup of the day
But basically they are both correct and you will be understood whichever you use.
The example "This car of John's..." is incorrect (see Janus comment).
A better example might be:
All the parents were invited to the school play, even the father of John and Anne, although he had been banned from attending on previous occasions
This avoids the difficulty, which confuses lots of people, of whether to say:
John's and Anne's father or John and Anne's father